Background: Pregnant women are said to have higher stress levels than non-pregnant women, but as non US studies have shown, stress increased during the pandemic due to the unique circumstances of unpredictability, fear of infection, limited access to health services, and financial uncertainty. The aim of this study was to determine the extent of stress associated with the coronavirus disease 2019 (COVID-19) pandemic among pregnant women in Slovenia and to determine in which areas they experienced the greatest stress. Methods: A descriptive and causal non-experimental method of empirical research was used. The research approach was based on quantitative research, and a validated questionnaire—Pandemic-Related Pregnancy Stress Scale (PREPS) was used as the research instrument. Eleven hundred and four pregnant women participated in the study. Results: We found that more than one-third of the participants experienced high levels of stress during the COVID-19 pandemic. The data demonstrated that those who were pregnant for the first time, had a high-risk pregnancy, experienced income loss, and were in their second or third trimester had higher levels of stress. The highest stress level was reported due to concern about infection. More specifically, 54.4% of pregnant women experienced high levels of stress related to fear of infection and in 47% high stress levels were related to fear with regard to childbirth. Conclusions: Respondents reported a range of mild to severe stress. They worried about the baby, about the need for isolation during labor and delivery, and about losing their social network in the postpartum period. It is important to know that stress during pregnancy also affects the fetus. We suggest that it would be useful to screen pregnant women for stress, with PREPS being used for this purpose. All women with high stress levels should be offered interventions, e.g., online stress reduction counselling.